Signals and Systems

5.5 The Multiplication Property

Imron Rosyadi

The Multiplication Property

If \(y[n]\) is the product of two signals, \(x_1[n]\) and \(x_2[n]\):

\[ y[n] = x_1[n] x_2[n] \]

Then their corresponding Fourier transforms are related by:

\[ Y\left(e^{j \omega}\right)=\frac{1}{2 \pi} \int_{2 \pi} X_{1}\left(e^{j \theta}\right) X_{2}\left(e^{j(\omega-\theta)}\right) d \theta \quad \text{(Eq. 5.63)} \]

• Multiplication in the time domain becomes periodic convolution in the frequency domain.
• The integral is performed over any interval of length \(2\pi\).
• This is analogous to the continuous-time multiplication property, but with periodic convolution due to the periodicity of the DTFT.

Important

This property is fundamental to understanding modulation, multiplexing, and spectral shaping in communication systems.

The multiplication property is the dual of the convolution property. While convolution in time becomes multiplication in frequency, multiplication in time becomes convolution in frequency. However, because the DTFT is periodic, this convolution in the frequency domain is a periodic convolution, meaning the integration is performed over a \(2\pi\) interval. This property is incredibly important in applications like communication systems, where signals are often multiplied by carrier waves for modulation, or in spectral analysis where windowing functions are multiplied with signals.

Derivation of the Multiplication Property

Let \(y[n] = x_1[n] x_2[n]\).

The DTFT of \(y[n]\) is:

\[ Y\left(e^{j \omega}\right)=\sum_{n=-\infty}^{+\infty} x_{1}[n] x_{2}[n] e^{-j \omega n} \]

Substitute the inverse DTFT of \(x_1[n]\):

\[ x_{1}[n]=\frac{1}{2 \pi} \int_{2 \pi} X_{1}\left(e^{j \theta}\right) e^{j \theta n} d \theta \quad \text{(Eq. 5.60)} \]

So,

\[ Y\left(e^{j \omega}\right)=\sum_{n=-\infty}^{+\infty} x_{2}[n]\left\{\frac{1}{2 \pi} \int_{2 \pi} X_{1}\left(e^{j \theta}\right) e^{j \theta n} d \theta\right\} e^{-j \omega n} \quad \text{(Eq. 5.61)} \]

Interchange summation and integration:

\[ Y\left(e^{j \omega}\right)=\frac{1}{2 \pi} \int_{2 \pi} X_{1}\left(e^{j \theta}\right)\left[\sum_{n=-\infty}^{+\infty} x_{2}[n] e^{-j(\omega-\theta) n}\right] d \theta \quad \text{(Eq. 5.62)} \]

The bracketed term is \(X_2(e^{j(\omega-\theta)})\), leading to:

\[ Y\left(e^{j \omega}\right)=\frac{1}{2 \pi} \int_{2 \pi} X_{1}\left(e^{j \theta}\right) X_{2}\left(e^{j(\omega-\theta)}\right) d \theta \quad \text{(Eq. 5.63)} \]

The derivation of the multiplication property is straightforward. We start with the definition of the DTFT of the product signal \(y[n]\). Then, we substitute the inverse DTFT expression for one of the signals, say \(x_1[n]\). By interchanging the order of summation and integration, we can recognize the inner summation as the DTFT of \(x_2[n]\) evaluated at a shifted frequency \((\omega-\theta)\). This directly leads us to the periodic convolution integral. The steps are quite similar to the continuous-time derivation, with the key difference being the \(2\pi\) integration interval due to the periodicity of the DTFT.

Periodic vs. Aperiodic Convolution

• Aperiodic Convolution (standard):

  \(\int_{-\infty}^{\infty} A(\tau) B(t-\tau) d\tau\)

  The result extends over the combined support of \(A\) and \(B\).

• Periodic Convolution (for DTFT):

  \(\frac{1}{2 \pi} \int_{2 \pi} X_{1}\left(e^{j \theta}\right) X_{2}\left(e^{j(\omega-\theta)}\right) d \theta\)

  The integration is limited to a \(2\pi\) interval.

  Because \(X_1(e^{j\omega})\) and \(X_2(e^{j\omega})\) are \(2\pi\)-periodic, the result \(Y(e^{j\omega})\) will also be \(2\pi\)-periodic.

Note

To evaluate periodic convolution using standard aperiodic convolution, we can define one of the functions to be zero outside a single \(2\pi\) period. This is often done for visualization and calculation.

It’s important to distinguish between aperiodic (or linear) convolution and periodic convolution. Aperiodic convolution typically involves integrating over the entire real line, and the result’s duration is the sum of the durations of the convolved signals. Periodic convolution, as seen in the DTFT multiplication property, involves integration over a finite interval, specifically \(2\pi\). Since the DTFTs are periodic themselves, their periodic convolution also results in a periodic function. For practical calculation, we can often treat one of the functions as being zero outside a single \(2\pi\) period and then perform a standard linear convolution, which effectively gives us one period of the periodic convolution result.

Example 5.15: Product of Two Sinc Signals

Find the Fourier transform \(X(e^{j\omega})\) of \(x[n]=x_1[n] x_2[n]\), where:

\[ x_{1}[n]=\frac{\sin (3 \pi n / 4)}{\pi n} \quad \text{and} \quad x_{2}[n]=\frac{\sin (\pi n / 2)}{\pi n} \]

1. Find the DTFTs of \(x_1[n]\) and \(x_2[n]\): These are ideal lowpass filters (from Example 5.12, but scaled).

• \(X_1(e^{j\omega})\) is a rectangular pulse with cutoff \(3\pi/4\).
• \(X_2(e^{j\omega})\) is a rectangular pulse with cutoff \(\pi/2\).

Figure 5.19: One period of \(X_1(e^{j\omega})\) (shaded) and \(X_2(e^{j\omega})\) (solid).

Example 5.15: Product of Two Sinc Signals

2. Apply the multiplication property (periodic convolution):

\[ X\left(e^{j \omega}\right)=\frac{1}{2 \pi} \int_{-\pi}^{\pi} X_{1}\left(e^{j \theta}\right) X_{2}\left(e^{j(\omega-\theta)}\right) d \theta \quad \text{(Eq. 5.64)} \]

To visualize this, we can consider the aperiodic convolution of \(\hat{X}_1(e^{j\omega})\) (one period of \(X_1(e^{j\omega})\)) with \(X_2(e^{j\omega})\).

3. Result of the periodic convolution:

Figure 5.20: Result of the periodic convolution in Example 5.15.

The convolution of two rectangular pulses in the frequency domain yields a trapezoidal shape.

This example applies the multiplication property to a concrete problem: finding the DTFT of the product of two sinc functions. We know that sinc functions in time correspond to rectangular pulses in the frequency domain. So, multiplying two sincs in time means periodically convolving two rectangular pulses in frequency. The figure shows the two rectangular pulses. When you convolve two rectangular pulses, the result is typically a trapezoidal shape. The final figure illustrates this trapezoidal shape, which is the DTFT of the product signal. This example demonstrates how the multiplication property helps us understand the spectral characteristics of complex signals formed by multiplying simpler ones.

Interactive Demo: Periodic Convolution

Visualize the periodic convolution of two rectangular pulses in the frequency domain.

viewof W1_val = Inputs.range([0.1, 3.14], {value: 3.14/2, step: 0.1, label: "Width of $X_1(e^{j\omega})$ (cutoff $\omega_{c1}$)"});
viewof W2_val = Inputs.range([0.1, 3.14], {value: 3.14/4, step: 0.1, label: "Width of $X_2(e^{j\omega})$ (cutoff $\omega_{c2}$)"});

This interactive demo allows you to manipulate the widths of two rectangular pulses in the frequency domain, representing \(X_1(e^{j\omega})\) and \(X_2(e^{j\omega})\). The demo then calculates and displays the result of their periodic convolution, \(Y(e^{j\omega})\). As you change the widths, observe how the shape of the convolution result changes. You should see how the convolution of two rectangular pulses typically results in a trapezoidal shape, and how the width of this trapezoid relates to the individual widths of the pulses. This is a direct visualization of the effect of multiplying two signals in the time domain on their frequency spectra.

ECE Applications of the Multiplication Property

• Amplitude Modulation (AM):
  • Multiplying a message signal \(x[n]\) by a high-frequency carrier \(c[n]=\cos(\omega_c n)\) shifts the message spectrum to \(\pm \omega_c\).
  • This allows for efficient transmission over a channel.
• Windowing:
  • Multiplying a long signal (or an infinite one) by a finite-duration window function \(w[n]\) (e.g., rectangular, Hamming) extracts a segment of the signal.
  • In the frequency domain, this corresponds to convolving the signal’s spectrum with the window’s spectrum, which can introduce spectral leakage.
• Frequency-Division Multiplexing (FDM):
  • Multiple signals are modulated onto different carrier frequencies, allowing them to share the same physical channel without interfering.
  • The multiplication property explains how each signal’s spectrum is shifted to its assigned frequency band.

The multiplication property is not just a mathematical curiosity; it has profound implications and applications in Electrical and Computer Engineering. Amplitude modulation, the basis of many radio and communication systems, directly relies on this property. Multiplying a message signal by a carrier wave shifts its spectrum to the carrier frequency, enabling transmission. Windowing, a common technique in spectral analysis, also uses this property. When we multiply a signal by a finite-duration window, we are effectively convolving the signal’s true spectrum with the window’s spectrum, which can lead to spectral smearing. Finally, Frequency-Division Multiplexing, allowing multiple users or channels to share the same medium, is another direct application of shifting spectra through multiplication.

5.6 TABLES OF FOURIER TRANSFORM PROPERTIES AND BASIC FOURIER TRANSFORM PAIRS

TABLE 5.1 PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM

Property	Aperiodic Signal	Fourier Transform
	\(x[n]\)	\(\left.X\left(e^{j \omega}\right)\right]\) periodic with
	\(y[n]\)	\(Y\left(e^{j \omega}\right) \int\) period \(2 \pi\)
Linearity	\(a x[n]+b y[n]\)	\(a X\left(e^{j \omega}\right)+b Y\left(e^{j \omega}\right)\)
Time Shifting	\(x\left[n-n_{0}\right]\)	\(e^{-j \omega n_{0}} X\left(e^{j \omega}\right)\)
Frequency Shifting	\(e^{j \omega_{1 j} n} x[n]\)	\(X\left(e^{j\left(\omega-\omega_{01}\right)}\right)\)
Conjugation	\(x^{*}[n]\)	\(X^{*}\left(e^{-j \omega}\right)\)
Time Reversal	\(x[-n]\)	\(X\left(e^{j \omega}\right)\)
Time Expansion	\(x_{(k)}[n]= \begin{cases}x[n / k], & \text { if } n=\text { multiple of } k \\ 0, & \text { if } n \neq \text { multiple of } k\end{cases}\)	\(X\left(e^{j k \omega}\right)\)
Convolution	\(x[n] * y[n]\)	\(X\left(e^{j \omega}\right) Y\left(e^{j \omega}\right)\)
Multiplication	\(x[n] y[n]\)	\(\frac{1}{2 \pi} \int_{2 \pi} X\left(e^{j \theta}\right) Y\left(e^{j(\omega-\theta)}\right) d \theta\)
Differencing in Time	\(x[n]-x[n-1]\)	\(\left(1-e^{-j \omega}\right) X\left(e^{j \omega}\right)\)
Accumulation	\(\sum_{k=-\infty}^{n} x[k]\)	\(\frac{1}{1-e^{-j \omega}} X\left(e^{j \omega}\right)\)
Differentiation in Frequency	\(n x[n]\)	\(+\pi X\left(e^{j 0}\right) \sum_{k=-\infty}^{+\infty} \delta(\omega-2 \pi k)\) \(j \frac{d X\left(e^{j \omega}\right)}{d \omega}\)
Conjugate Symmetry for Real Signals	\(x[n]\) real	\(\left\{\begin{array}{l}X\left(e^{j \omega}\right)=X^{*}\left(e^{-j \omega}\right) \\ \operatorname{Re}_{e}\left\{X\left(e^{j \omega}\right)\right\}=\operatorname{Re}\left\{X\left(e^{-j \omega}\right)\right\} \\ \mathfrak{I} m_{b}\left\{X\left(e^{j \omega}\right)\right\}=-\mathfrak{I} n_{\mathfrak{b}}\left\{X\left(e^{-j \omega}\right)\right\} \\ \left\|X\left(e^{j \omega}\right)\right\|=\left\|X\left(e^{-j \omega}\right)\right\| \\ \angle X\left(e^{j \omega}\right)=-\angle X\left(e^{-j \omega}\right)\end{array}\right.\)
Symmetry for Real, Even Signals	\(x[n]\) real an even	\(X\left(e^{j \omega}\right)\) real and even
Symmetry for Real, Odd Signals	\(x[n]\) real and odd	\(X\left(e^{j \omega}\right)\) purely imaginary and \(\quad\) odd
Even-odd Decomposition	\(x_{e}[n]=\mathcal{E}_{v}\{x[n]\} \quad[x[n]\) real \(]\)	\(\operatorname{Re}\left\{X\left(e^{j \omega}\right)\right\}\)
of Real Signals	\(x_{o}[n]=\operatorname{Od}\{x[n]\} \quad[x[n]\) real \(]\)	\(j \mathscr{I} m\left\{X\left(e^{j \omega}\right)\right\}\)
Parseval’s R \(\sum_{n=-\infty}^{+\infty} \mid x[n]\)	ation for Aperiodic Signals \(=\frac{1}{2 \pi} \int_{2 \pi}\left\|X\left(e^{j \omega}\right)\right\|^{2} d \omega\)

5.6 TABLES OF FOURIER TRANSFORM PROPERTIES AND BASIC FOURIER TRANSFORM PAIRS

TABLE 5.2 BASIC DISCRETE-TIME FOURIER TRANSFORM PAIRS

Signal	Fourier Transform	Fourier Series Coefficients (if periodic)
\(\sum_{k=\langle N\rangle} a_{k} e^{j k(2 n / N) n}\)	\(2 \pi \sum_{k=-\infty}^{+\infty} a_{k} \delta\left(\omega-\frac{2 \pi k}{N}\right)\)	\(a_{k}\)
\(e^{j \omega_{0} n}\)	\(2 \pi \sum_{l=-\infty}^{+\infty} \delta\left(\omega-\omega_{0}-2 \pi l\right)\)
\(\cos \omega_{0} n\)	\(\pi \sum_{l=-\infty}^{+\infty}\left\{\delta\left(\omega-\omega_{0}-2 \pi l\right)+\delta\left(\omega+\omega_{0}-2 \pi l\right)\right\}\)	(a) \(\omega_{0}=\frac{2 \pi m}{N}\) \(a_{k}= \begin{cases}\frac{1}{2}, & k= \pm m, \pm m \pm N, \pm m \pm 2 N, \ldots \\ 0, & \text { otherwise }\end{cases}\) (b) \(\frac{\omega_{0}}{2 \pi} \quad\) irrational \(\Rightarrow\) The signal is aperiodic
\(\sin \omega_{0} n\)	\(\frac{\pi}{j} \sum_{l=-\infty}^{+\infty}\left\{\delta\left(\omega-\omega_{0}-2 \pi l\right)-\delta\left(\omega+\omega_{0}-2 \pi l\right)\right\}\)	(a) \(\quad \omega_{0}=\frac{2 \pi r}{N}\) \(\quad a_{k}=\left\{\begin{aligned} \frac{1}{2 j}, & k=r, r \pm N, r \pm 2 N, \ldots \\ -\frac{1}{2 j}, & k=-r,-r \pm N,-r \pm 2 N, \ldots \\ 0, & \text { otherwise }\end{aligned}\right.\) (b) \(\frac{\omega_{0}}{2 \pi} \quad\) irrational \(\Rightarrow\) The signal is aperiodic
\(x[n]=1\)	\(2 \pi \sum_{l=-\infty}^{+\infty} \delta(\omega-2 \pi l)\)	\(a_{k}= \begin{cases}1, & k=0, \pm N, \pm 2 N, \ldots \\ 0, & \text { otherwise }\end{cases}\)
Periodic square wave \(x[n]= \begin{cases}1, & \|n\| \leq N_{1} \\ 0, & N_{1}<\|n\| \leq N / 2\end{cases}\) and \(x[n+N]=x[n]\)	\(2 \pi \sum_{k=-\infty}^{+\infty} a_{k} \delta\left(\omega-\frac{2 \pi k}{N}\right)\)	\(a_{k}=\frac{\sin \left[(2 \pi k / N)\left(N_{1}+\frac{1}{2}\right)\right]}{N \sin [2 \pi k / 2 N]}, k \neq 0, \pm N, \pm 2 N, \ldots\) \(a_{k}=\frac{2 N_{1}+1}{N}, k=0, \pm N, \pm 2 N, \ldots\)
\(\sum_{k=-\infty}^{+\infty} \delta[n-k N]\)	\(\frac{2 \pi}{N} \sum_{k=-\infty}^{+\infty} \delta\left(\omega-\frac{2 \pi k}{N}\right)\)	\(a_{k}=\frac{1}{N}\) for all \(k\)
\(a^{n} u[n], \quad\|a\|<1\)	\(\frac{1}{1-a e^{-j \omega}}\)	-
\(x[n] \begin{cases}1, & \|n\| \leq N_{1} \\ 0, & \|n\|>N_{1}\end{cases}\)	\(\frac{\sin \left[\omega\left(N_{1}+\frac{1}{2}\right)\right]}{\sin (\omega / 2)}\)	-
\(\frac{\sin W n}{\pi n}=\frac{W}{\pi} \operatorname{sinc}\left(\frac{W n}{\pi}\right)\) \(0<W<\pi\)	\(X(\omega)= \begin{cases}1, & 0 \leq\|\omega\| \leq W \\ 0, & W<\|\omega\| \leq \pi\end{cases}\) \(X(\omega)\) periodic with period \(2 \pi\)	-
\(\delta[n]\)	1	-
\(u[n]\)	\(\frac{1}{1-e^{-j \omega}}+\sum_{k=-\infty}^{+\infty} \pi \delta(\omega-2 \pi k)\)	-
\(\delta\left[n-n_{0}\right]\)	\(e^{-j \omega n_{0}}\)	-
\((n+1) a^{n} u[n], \quad\|a\|<1\)	\(\frac{1}{\left(1-a e^{-j \omega}\right)^{2}}\)	-
\(\frac{(n+r-1) !}{n !(r-1) !} a^{n} u[n], \quad\|a\|<1\)	\(\frac{1}{\left(1-a e^{-j \omega}\right)^{r}}\)	-